Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,5)(6,8)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24)(25,26,30,28)(27,31,33,34)(29,36,37,32)(35,40,39,38)(41,42)(43,49)(44,52)(45,48)(46,51)(47,54)(50,55)(53,56);(10,14)(11,18)(16,20)(19,24)(27,33)(31,34)(35,39)(38,40)(41,42,45,48)(43,46,51,49)(44,47,54,52)(50,55,53,56);(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,23)(21,24)(25,27,30,33)(26,31,28,34)(29,35,37,39)(32,38,36,40)(41,42)(43,46)(44,47)(45,48)(49,51)(50,55)(52,54)(53,56);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,20)(13,14)(16,21)(17,18)(19,23)(22,24)(25,28,30,26)(27,34,33,31)(29,36,37,32)(35,40,39,38)(41,43,45,51)(42,46,48,49)(44,53,54,50)(47,56,52,55);(1,3,4,7)(2,6,5,8)(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,29,30,37)(26,32,28,36)(27,35,33,39)(31,38,34,40)(41,44,45,54)(42,47,48,52)(43,50,51,53)(46,55,49,56)Polycyclic presentation:
2^9|a^2=fh,b^2=fg,c^2=fh,d^2=gh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4819451);